Flirc basically works like a keyboard. You teach it by selecting a keyboard key or key combination (with modifiers) and assign it to a button on a remote, by pressing the button. It should work with most remotes but some may cause problems. Flirc is "tuned" to remotes which use 38 kHz carrier frequency. This frequency is used by most consumer appliances' remotes. It won't work properly with remotes which use frequencies too far from that, like some MCE remotes working at 50-something kHz for example.
Also there are many different IR protocols. Some work better than others and some may be unusable. For example many settop boxes (like cable boxes) use strange proprietary protocols that don't work well with Flirc.
I think the best solution is to look for a universal remote that lay good in your hand. You can then switch device profiles and test which works best for you. My personal preference is to use a device profile which uses one of the NEC family protocols (preferably NECx1 or NECx2). For example LG uses NEC protocols (NEC1 for TVs, NECx1 for soundbars/sound systems). Also Samsung uses NEC for some TVs. I don't know what Panasonic uses but @jason also recommends that brand. I don't recommend Philips or MCE as these are RC5/RC6 based. These protocols have toggle bit that changes on each key press. This means you'd need to record each remote button twice.
I'm using AllForOne Simple 4 remote (URC-6440). It's really nice universal remote and the price is really good. It's 4-device by default, but there's a possibility to upgrade it to 12-device one using custom firmware (really easy to install and fully reversible). The custom firmware also adds a lot of features. There's also a community developed PC software (for Windows, Mac and Linux!) that allows you to configure the remote whatever you want (the remote has microUSB port to connect to PC). If you can learn something about this and other remotes from the same manufacturer, you can go to http://hifi-remote.com/forums/. It's a community forum focused on providing support and extra features for this remotes.
You can also go the Logitech Harmony way. There're even ready to use Flirc profiles in the Harmony software which are supported by Flirc without any programming. They do not give you 100% of functionality out of a box. Rather something like a quick start. I don't have Harmony so I can't tell you much about them.